Sotah, Chapter 5, Mishnah 4. This mishnah too was taught "on that day," and it comes to discuss exactly how the Song of the Sea was recited by Moshe and the Jewish people. Along the way we will also learn about the way Hallel was recited in the time of the Gemara.
Rabbi Akiva's Derashah:
The mishnah opens: "Bo ba-yom darash Rabbi Akiva" - on that day Rabbi Akiva expounded. Rabbi Akiva expounds the verse "Then Moshe and the children of Israel sang this song to Hashem, and they said, saying." On the face of it the word "saying" is superfluous - they already sang the song, so what does the word add? On this the mishnah asks: "U-mah talmud lomar le-emor?" - what does the word "saying" come to teach? And it answers: "Melamed she-hayu Yisrael onin acharav shel Moshe al kol davar ve-davar ke-korin es ha-Hallel" - it teaches that the Jewish people responded after Moshe for each and every phrase, the way Hallel is recited. The seemingly extra word teaches that there was a response from the Jewish people, who answered after Moshe for every expression and every single phrase that he said.
How the Leader Recited Hallel in the Time of the Gemara:
The Gemara explains that the Jewish people recited the song "like an adult leading the recitation of Hallel." In the time of the Gemara, the adult leader would fulfill the obligation on behalf of the entire congregation: he intended to fulfill their obligation, and they would listen and respond to his words.
In the recitation of Hallel: The leader would say "Halelu avdei Hashem," and the congregation would answer "Halleluyah." So too in the second paragraph: when he reached "Be-tzeis Yisrael mi-Mitzrayim," the congregation repeated those same words after him, and from that point on they answered "Halleluyah" after each and every sentence.
In the Song of the Sea: Moshe said "Ashirah la-Hashem," and they answered after him "Ashirah la-Hashem"; he said the next words, "ki ga'oh ga'ah," and again they answered "Ashirah la-Hashem." This was something like saying amen, in the manner in which they would recite Hallel.
This, then, is the meaning of the word "saying."
The View of Rabbi Nechemiah:
Rabbi Nechemiah disagrees and holds that the song was recited after leaving the sea "ke-korei es Shema" - like one reciting Shema, and not the way Hallel is recited, meaning that everyone said it together in unison. The Gemara explains that his words come to exclude both ways in which Hallel is led: not the way an adult leads it, and not the way a child leads it either.
For there is an opinion in the Gemara - which Rabbi Nechemiah in our mishnah does not follow - that the song was recited in the manner in which a child leads Hallel. When a child leads, he cannot fulfill the obligation on behalf of the congregation, and therefore the congregation repeats after him: the child says a verse, and the congregation repeats that same verse after him. In this way the congregation says the words itself, except that it is repeating after the one leading.
According to Rabbi Nechemiah, they recited the song like the recitation of Shema, all of it together and in a single voice. This required that the entire Jewish people be infused with that same divine spirit, so that they all knew together what to say. This is almost a form of prophecy - not actual prophecy, but something that directed them all to say the same words at the same time.
In summary: The mishnah expounds the word "saying" in the verse "Then Moshe sang." According to Rabbi Akiva it teaches that the Jewish people responded after Moshe for each and every phrase, the way an adult leads the recitation of Hallel and thereby fulfills the congregation's obligation; and according to Rabbi Nechemiah the song was recited like the recitation of Shema, in one voice and in unison, and not in the manner in which Hallel is led - neither by an adult nor by a child.